CompositionMajorIndex

The sum of the descents of the composition: its partial sums, the total left out.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000008via IntegerComposition✓↗

CompositionMajorIndex(composition)The sum of the descents of the composition: its partial sums, the total left out.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• Reads the composition as a descent set, as FindStat's St000008 does. MajorIndex reads it as a word.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 2 else Sum(Map(
                                  Function(Sum(Map(
                                                    Function(_x[j], j),
                                                    Range(1, i),
                                                  )), i),
                                  Range(1, Length(_x) - 1),
                                ))

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{CompositionMajorIndex}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$18$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([1]), \text{CompositionMajorIndex})$Out$0$

Known value · FindStat St000008

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{CompositionMajorIndex})$Out$0$

Known value · FindStat St000008

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{CompositionMajorIndex})$Out$0$

Known value · FindStat St000008

About

Examples	4
Crosswalk	FindStat